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Journal of Nonlinear Mathematical Physics Volume 9, 2002 - Issue sup1: Recent Advances in Integrable Systems
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Original Articles

Is My ODE a Painlevé Equation in Disguise?

Jarmo Hietarinta Department of Physics, University of Turku, FIN-20014, Turku, Finland E-mail: [email protected]
&
Valery Dryuma Institute of Mathematics AS RM, Academy str. 5, 277028, Kishinev, Moldova E-mail: [email protected]
Pages 67-74 | Received 08 May 2002, Accepted 19 Oct 2001, Published online: 21 Jan 2013
 
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Abstract

Painlevé equations belong to the class y’’+a 1 y’ 3+3a 2 y’ 2+3a 3 y’+a 4=0, where a i=a i(x, y). This class of equations is invariant under the general point transformation x=Φ(X, Y ), y=Ψ(X, Y ) and it is therefore very difficult to find out whether two equations in this class are related. We describe R. Liouville’s theory of invariants that can be used to construct invariant characteristic expressions (syzygies), and in particular present such a characterization for Painlevé equations I-IV.

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